Chapter 28, Problem 28.23QAP

Interpretation Introduction

(a)

Interpretation:

The optimal velocity is to be stated.

Concept introduction:

The Van Deemeter equation relates the HPLC plate height and velocity of plate. The Van Deemeter equation is as follows.

$\begin{array}{c}H=\frac{\text{B}}{u}+\text{C}u\\ =\frac{\text{B}}{u}+{\text{C}}_{\text{S}}u+{\text{C}}_{\text{M}}u\end{array}$ ...... (I)

Here, the HPLC height is $H$, the plate velocity is $u$, the diffusion coefficient is $\text{B}$, the mass transfer coefficient in stationary phase is ${\text{C}}_{\text{S}}$, the mass transfer coefficient in mobile phase is ${\text{C}}_{\text{M}}$, the total mass transfer coefficient is $\text{C}$.

Answer to Problem 28.23QAP

The optimal velocity of HPLC plate is $\underset{\_}{\sqrt{\frac{\text{B}}{\text{C}}}}$.

Explanation of Solution

Differentiate Equation (I) with respect to plate velocity.

$\frac{dH}{du}=\frac{\left(-1\right)\text{B}}{u^2}+\text{C}\left(1\right)$ ...... (II)

Substitute $0$ for $\frac{dH}{du}$, $u_{opt}$ for $u$ in the equation for optimal velocity.

$\begin{array}{c}0=\frac{-\text{B}}{u_{opt}^2}+\text{C}\\ u_{opt}=\sqrt{\frac{\text{B}}{\text{C}}}\end{array}$

Interpretation Introduction

(b)

Interpretation:

The minimum plate height is to be stated.

Concept introduction:

The Van Deemeter equation relates the HPLC plate height and velocity of plate. The Van Deemeter equation is as follows.

$\begin{array}{c}H=\frac{\text{B}}{u}+\text{C}u\\ =\frac{\text{B}}{u}+{\text{C}}_{\text{S}}u+{\text{C}}_{\text{M}}u\end{array}$

The relation between HPLC height is inversely proportional to plate velocity. So minimum height is obtained corresponding to optimal velocity in the Van Deemeter equation.

Answer to Problem 28.23QAP

The minimum plate height is $\underset{\_}{2\sqrt{\text{BC}}}$.

Explanation of Solution

Substitute $\sqrt{\frac{\text{B}}{\text{C}}}$ for $u$ in the Equation (I).

$\begin{array}{c}{H}_{\min }=\frac{\text{B}}{\left(\sqrt{\frac{\text{B}}{\text{C}}}\right)}+\text{C}\sqrt{\frac{\text{B}}{\text{C}}}\\ =\sqrt{\text{BC}}+\sqrt{\text{BC}}\\ =2\sqrt{\text{BC}}\end{array}$

Interpretation Introduction

(c)

Interpretation:

The optimal velocity and minimum height of plate in terms of the diffusion coefficient in mobile phase, particle size and dimensional analysis is to be stated.

Concept introduction:

The mass transfer coefficient relates the particle size and diffusion coefficient in mobile phase. The diffusion coefficient relates the dimensional constant and diffusion coefficient in mobile phase.

Answer to Problem 28.23QAP

The optimal velocity is $\underset{\_}{\frac{{\text{D}}_{\text{M}}}{d_P}\sqrt{\frac{2\gamma }{\omega }}}$ and the minimum height is $\underset{\_}{2d_P\sqrt{2\gamma \omega }}$ of plate in terms of the diffusion coefficient in mobile phase, particle size and dimensional analysis is to be stated.

Explanation of Solution

Write the expression for mass transfer coefficient in mobile phase.

${\text{C}}_{\text{M}}=\frac{\omega d_p^2}{{\text{D}}_{\text{M}}}$ ...... (III)

Here, the diffusion coefficient in mobile phase is ${\text{D}}_{\text{M}}$, the particle size is $d_P$, the dimensional constant is $\omega$.

Write the expression for diffusion coefficient.

$\text{B}=2\gamma {\text{D}}_{\text{M}}$ ...... (IV)

Here, the dimensional coefficient is $\gamma$.

Write the expression for optimal velocity in mobile phase.

$u_{opt}=\sqrt{\frac{\text{B}}{{\text{C}}_{\text{M}}}}$ ...... (V)

Write the expression for minimum height in mobile phase.

$H_{\min }=2\sqrt{{\text{BC}}_{\text{M}}}$ ...... (VI)

Substitute $\frac{\omega d_p^2}{{\text{D}}_{\text{M}}}$ for ${\text{C}}_{\text{M}}$, $2\gamma {\text{D}}_{\text{M}}$ for $\text{B}$ in the Equation (V).

$\begin{array}{c}{u}_{opt}=\sqrt{\frac{\left(2\gamma {\text{D}}_{\text{M}}\right)}{\left(\frac{\omega d_p^2}{{\text{D}}_{\text{M}}}\right)}}\\ =\sqrt{\frac{2\gamma {\text{D}}_{\text{M}}^2}{\omega d_p^2}}\\ =\frac{{\text{D}}_{\text{M}}}{d_P}\sqrt{\frac{2\gamma }{\omega }}\end{array}$ ...... (VII)

Substitute $\frac{\omega d_p^2}{{\text{D}}_{\text{M}}}$ for ${\text{C}}_{\text{M}}$, $2\gamma {\text{D}}_{\text{M}}$ for $\text{B}$ in the Equation (VI).

$\begin{array}{c}{H}_{\min }=2\sqrt{\left(2\gamma {\text{D}}_{\text{M}}\right)\left(\frac{\omega d_p^2}{{\text{D}}_{\text{M}}}\right)}\\ =2d_P\sqrt{2\gamma \omega }\end{array}$ ...... (VIII)

Interpretation Introduction

(d)

Interpretation:

The optimal velocity and minimum height of plate at unity order for dimensional constants.

Concept introduction:

The mass transfer coefficient relates the particle size and diffusion coefficient in mobile phase. The diffusion coefficient relates the dimensional constant and diffusion coefficient in mobile phase.

Answer to Problem 28.23QAP

The optimal velocity is $\underset{\_}{\frac{{\text{D}}_{\text{M}}}{d_P}}$ and minimum height of plate is $\underset{\_}{d_P}$ at unity order for dimensional constants.

Explanation of Solution

Write the expression for optimal velocity in terms of dimensional constants.

$u_{opt}=\frac{{\text{D}}_{\text{M}}}{d_P}\sqrt{\frac{2\gamma }{\omega }}$ ...... (IX)

Substitute $1$ for $\gamma$, and $1$ for $\omega$ in the Equation (IX).

$\begin{array}{c}{u}_{opt}=\frac{{\text{D}}_{\text{M}}}{d_P}\sqrt{\frac{2\left(1\right)}{\left(1\right)}}\\ =\frac{\sqrt{2}{\text{D}}_{\text{M}}}{d_P}\\ \approx \frac{{\text{D}}_{\text{M}}}{d_P}\end{array}$

Write the expression for plate height in terms of dimensional constants.

$H_{\min }=2d_P\sqrt{2\gamma \omega }$ ...... (X)

Substitute $1$ for $\gamma$, and $1$ for $\omega$ in the Equation (X).

$\begin{array}{c}{H}_{\min }=2d_P\sqrt{2\left(1\right)\left(1\right)}\\ =2\sqrt{2}{d}_P\\ \approx d_P\end{array}$

Interpretation Introduction

(e)

Interpretation:

The preceding conditions for reduced plate height to one third, the optimal velocity in these conditions and effect on number of theoretical plates is to be stated.

Concept introduction:

The plate height and number of theoretical plates directly depends on the particle size and optimal velocity inversely depends on particle size.

Answer to Problem 28.23QAP

For one third reduction in particle size the plate height is reduced to one third. The optimal velocity becomes $\frac{3}{2}$ times to initial optimal velocity. The number of theoretical plates also reduced to one third of initial value.

Explanation of Solution

Write the expression for plate height.

$\frac{H_1}{d_{p1}}=\frac{H_2}{d_{p2}}$ ...... (XII)

Here, the initial plate height is $H_1$, reduced plate height is $H_2$, the initial particle size is $d_{p1}$, the particle size at reduced plate height is $d_{p2}$.

Substitute $\frac{2}{3}{H}_1$ for $H_2$ in the equation (XII).

$\begin{array}{l}\frac{H_1}{d_{p1}}=\frac{\frac{2}{3}\left(H_1\right)}{d_{p2}}\\ d_{p2}=\frac{2}{3}{d}_{p1}\end{array}$

The particle size is also reduced to one third to initial particle size for one third reduction in the plate height.

The optimal velocity is inversely proportional to the particle size, so optimal velocity become $\frac{3}{2}$ times to initial optimal velocity.

Write the expression for number of theoretical plates.

$N=\frac{H}{L}$

From the above equation it is clear that number of theoretical plates reduced to one third to initial number of theoretical plates.

Interpretation Introduction

(f)

Interpretation:

The condition to reduce the plate height to one third without reducing the number of theoretical plates is to be stated.

Concept introduction:

The number of theoretical plates directly depends on the height of the plate and inversely to the column length.

Answer to Problem 28.23QAP

The column length is also reduced to one third with plate height to maintain same number of theoretical plates.

Explanation of Solution

Write the expression for number of theoretical plate with one third reduced plate height

$N=\frac{\frac{2}{3}H}{L}$ ...... (XIII)

To maintain the same number of theoretical plate length of column also be reduced to one third. We have to substitute $\frac{2}{3}$ in the equation (XIII).

Interpretation Introduction

(g)

Interpretation:

The two sources of band broadening which also contribute to overall width of HPLC peaks is to be stated.

Concept introduction:

The band broadening is defined as the reduction in separation of solute molecules in the column. This also reduces the quality and accuracy of the separation. The band broadening occurs due to longitudinal movement of particles in the column.

Answer to Problem 28.23QAP

The two sources of band broadening which also contribute to overall width of HPLC peaks are diameter of the particles and diffusion coefficient in mobile phase.

Explanation of Solution

The band broadening is the measure of column efficiency. It is inversely proportional to the mass transfer coefficient. If the mass transfer is slow in the column then wide band is obtained and narrow band is obtained when the mass transfer in the column is high. The sources of extra column band broadening which contributes to overall width of liquid chromatography peaks are following.

(1) The diameter of particles in column.

(2) The diffusion coefficient of mobile phase.